A problem related to Bárány Grünbaum conjecture

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1 Filoma 27:1 (2013), DOI /FIL B Published by Faculy of Sciences and Mahemaics, Universiy of Niš, Serbia Available a: hp:// A problem relaed o Bárány Grünbaum conjecure Pavle V. M. Blagojević a, Aleksandra Dimirijević Blagojević b a Mahemaički Insiu SANU, Knez Mihajlova 36, Beograd, Serbia b Mahemaički Insiu SANU, Knez Mihajlova 36, Beograd, Serbia Absrac. In his paper we prove ha for any absolue coninuous Borel probabiliy measure µ on he sphere S 2 and any [0, 1 4 ] here exis four grea semi-circles l 1,..., l 4 emanaing from a poin x S 2 ha pariion sphere S 2 ino four angular secors σ 1,..., σ 4, couner clockwise oriened, such ha µ(σ 1 ) = µ(σ 4 ) =, µ(σ 2 ) = µ(σ 3 ) = 1 4, and area(σ 1) = area(σ 4 ), area(σ 2 ) = area(σ 3 ). 1. Inroducion and saemen of resuls Le P be a convex body in he plane. I is know ha here exis wo orhogonal lines ha par P ino four pieces of equal area. The following naural quesion was asked joinly by Imre Bárány and Branko Grünbaum. Conjecure 1.1 (Bárány Grünbaum conjecure in R 2 ). Le P be a convex body in he plane of area 1 and [0, 1 4 ]. There exis wo orhogonal lines ha pariion P ino four pieces of area,, 1 2 and 1 2 in couner clockwise order. In he case when he diameer of P is a leas 37 imes he minimum widh, he conjecure is seled by Arocha, Jernimo-Casro, Monejano, and Roldn-Pensado in [1]. The conjecure can be naurally generalized o he following quesion: Le µ be an absoluely coninuous, Borel, probabliy measure on R 2 and [0, 1 4 ]. There exis wo orhogonal lines ha pariion R 2 ino four pieces ha conain,, 1 2 and 1 2 amoun of measure µ, in couner clockwise order. Moving he problem from he plane o he sphere S 2 we ge a similar, no equivalen, bu equally resisan conjecure. Conjecure 1.2 (Bárány Grünbaum conjecure on S 2 ). Le µ be an absoluely coninuous, Borel, probabliy measure on he sphere S 2 and [0, 1 4 ]. There exis wo grea circles l 1 and l 2 ha are muually orhogonal and pariion sphere S 2 ino four angular secors σ 1, σ 2, σ 3, σ 4, couner clockwise oriened, wih he propery ha µ(σ 1 ) = µ(σ 2 ) = µ(σ 3 ) = µ(σ 4 ) Mahemaics Subjec Classificaion. Primary 52A37; Secondary 55S91, 55R20 Keywords. Measure pariions, exisence of equivarian maps, Serre specral sequence Received: 29 May 2012; Acceped: 19 Sepember 2012 Communicaed by Ljubiša D.R. Kočinac The research of he firs auhor leading o hese resuls has received funding from he European Research Council under he European Union s Sevenh Framework Programme (FP7/ ) / ERC Gran agreemen no SDModels. Boh auhors are also suppored by he gran ON of he Serbian Minisry Educaion and Science. addresses: pavleb@mi.sanu.ac.rs (Pavle V. M. Blagojević), aleksandra1973@gmail.com (Aleksandra Dimirijević Blagojević)

2 P. Blagojević, A. Dimirijević Blagojević / Filoma 27:1 (2013), v 4 v 3 v x 1 v 2 l 1 Figure 1: An example of a 4-fan The grea circles l 1 and l 2 on he sphere S 2 are muually orhogonal if he uni angen vecors v 1 and v 2 o l 1 and l 2 a one of wo anipodal inersecion poins are orhogonal, i.e., v 1 v 2. In his paper we prove he following weaker version of he Bárány Grünbaum conjecure on S 2. Theorem 1.3. Le µ be an absoluely coninuous, Borel, probabiliy measure on he sphere S 2 and [0, 1 4 ]. There exis four grea semi-circles l 1,..., l 4 emanaing from a poin x S 2 ha pariion sphere S 2 ino four angular secors couner clockwise oriened: σ 1 = (l 1, l 2 ), σ 2 = (l 2, l 3 ), σ 3 = (l 3, l 4 ), σ 4 = (l 4, l 1 ), having propery ha µ(σ 1 ) = µ(σ 4 ) =, µ(σ 2 ) = µ(σ 3 ) = 1 4 and area(σ 1) = area(σ 4 ), area(σ 2 ) = area(σ 3 ). The condiion abou he areas can be reformulaed in erms of angles ha deermine angular secors. Le us denoe by α 1 = (l 1, l 2 ), α 2 = (l 2, l 3 ), α 3 = (l 3, l 4 ), α 4 = (l 4, l 1 ). Then area(σ i ) = α i 2π area(s2 ) for each i {1, 2, 3, 4}. Therefore, condiion on areas read off in erms of angles as α 1 = α 4 and α 2 = α 3, which implies ha l 1 l 3 is a grea circle, or l 1 = l From geomeric problem o an equivarian problem The proof of Theorem 1.3 is obained via he configuraion es map mehod. In his secion we relae he claim of he heorem wih he non-exisence of a Z/2-equivarian map from he Siefel manifold V 2 (R 3 ) ino he sphere S 1. A 4-fan (x; l 1,..., l 4 ) on he sphere S 2 consiss of a poin x S 2 on he sphere and four pairwise differen grea semi-circles l 1,..., l 4 emanaing from x oriened in he couner clockwise order. For a 4-fan (x; l 1,..., l 4 ) we can also use noaion 1. (x; σ 1,..., σ 4 ) where σ i denoes he open angular secor beween l i and l i+1, i {1,..., 4}, l 5 l 1 ; or 2. (x; v 1,..., v 4 ) where v i T x S 2 denoes he unie angen vecor deermined by he grea semicircle curve l i, i {1,..., 4}. Observe ha v 1,..., v 4 span(x). Furher on, F 4 denoes he space of all 4-fans on S 2. Le µ be an absoluely coninuous, Borel, probabliy measure on he sphere S 2 and [0, 1 4 ]. Consider he following configuraion space deermined by µ and give : X µ, := {(x; σ 1,..., σ 4 ) F 4 : µ(σ 1 ) = µ(σ 4 ) =, µ(σ 2 ) = µ(σ 3 ) = 1 4 }.

3 P. Blagojević, A. Dimirijević Blagojević / Filoma 27:1 (2013), Every poin (x; v 1,..., v 4 ) = (x; l 1,..., l 4 ) in he space X µ, is compleely deermined by he poin x and he firs angen vecor v 1 o l 1. Afer fixing (x, v 1 ) V 2 (R 3 ) he remaining angen vecors v 2, v 3, v 4 can be obain from he condiion µ(σ 1 ) = µ(σ 4 ) =, µ(σ 2 ) = µ(σ 3 ) = 1 4. Thus, X µ, V 2 (R 3 ). Moreover, he configuraion space X µ, has a naural free Z/2 = ε acion given by ε (x; l 1, l 2, l 3, l 4 ) = ( x; l 1, l 4, l 3, l 2 ) or ε (x; σ 1, σ 2, σ 3, σ 4 ) = ( x; σ 4, σ 3, σ 2, σ 1 ). Le us define a coninuous map f µ, : X R 2 by (x; σ 1, σ 2, σ 3, σ 4 ) (area(σ 1 ) area(σ 4 ), area(σ 2 ) area(σ 3 )). If we inroduce he anipodal Z/2-acion on R 2 i is no hard o see ha he map τ is a Z/2-equivarian map. Indeed, he following diagram commues (x; σ 1, σ 2, σ 3, σ 4 ) f µ, (area(σ 1 ) area(σ 4 ), area(σ 2 ) area(σ 3 )) ( x; σ 4, σ 3, σ 2, σ 1 ) ε f µ, (area(σ 4 ) area(σ 1 ), area(σ 3 ) area(σ 2 )). ε The main properies of his map, coming from is consrucion, are summarised in he following proposiion. Proposiion 2.1. (i) If for he given µ and he claim of Theorem 1.3 holds, hen 0 im f µ, R 2. (ii) If for he given he claim of Theorem 1.3 does no hold, hen here exiss a measure µ such ha 0 im f µ, R 2. Consequenly, he map f µ, facors in he following way f µ, V 2 (R 3 ) X µ, R 2 \{0} R 2 where µ, : V 2 (R 3 ) R 2 \{0} is a Z/2-equivarian map. (iii) If for he given he claim of Theorem 1.3 does no hold, hen here exiss a Z/2-equivarian map V 2 (R 3 ) S 1. The acion on S 1 is assumed o be anipodal. (iv) If here is no Z/2-equivarian map V 2 (R 3 ) S 1, hen he claim of Theorem 1.3 holds for every and any µ. Thus, if we prove ha here is no Z/2-equivarian map V 2 (R 3 ) S 1 we have concluded he proof of Theorem Non exisence of an equivarain map The non-exisence of a Z/2-equivarian map V 2 (R 3 ) S 1 will be proved via he Fadell Husseini ideal valued index heory applied o he group Z/2 and ineger coefficiens. We make a quick review of basic properies of he Fadell Husseini index heory. For furher deails consul he original paper of Fadell and Husseini [3] and for use of ineger coefficiens [2]. Consider a finie group G. Le X be a G-space, R a commuaive ring wih uni and p X : X p he G-equivarian projecion. The Fadell Husseini index of he G-space X is he kernel ideal of he induced map p X : H G (p; R) H (X; R) in he equivarian cohomology, i.e., G Index G (X; R) := ker ( p X : H G (p; R) H G (X; R)). Here H G (X; R) := H (EG G X; R) and herefore H G (p; R) H (G; R).

4 P. Blagojević, A. Dimirijević Blagojević / Filoma 27:1 (2013), Le X and Y be G-spaces and f : X Y a G-equivarian map. There are following commuaive riangles of G-spaces and relaed cohomology groups: X p X f p Y p Y f H G (X; R) H (Y; R) G p X p Y (p; R). H G Consequenly, ker p Y ker p X, i.e., Index G (Y; R) Index G (X; R) H G (p; R) = H (G; R). (1) Le us recall ha he cohomology ring of he group Z/2, or Z/2-equivarian cohomology of he poin, wih ineger coefficiens Z can be presened by H (Z/2; Z) = Z[T]/ 2T where deg(t) = 2. In compuaion of indexes ha follows we use he Serre specral sequence of he Borel consrucion fibraions: S 1 EZ/2 Z/2 S 1 BZ/2 and V 2 (R 3 ) EZ/2 Z/2 V 2 (R 3 ) BZ/2. The E 2 -erms, wih Z coefficiens, of hese specral sequences (S 1 ) and (V 2 (R 3 )) are given by 2 (S1 ) = H r (Z/2; H s (S 1 ; Z)) and 2 (V 2(R 3 )) = H r (Z/2; H s (V 2 (R 3 ); Z)) Index Z/2 (S 1 ; Z) The group Z/2 acs on he sphere S 1 anipodally and herefore orienaion preserving. Consequenly, H s (V 2 (R 3 ); Z) is a rivial Z/2-module and so he E 2 -erm of he (S 1 ) specral sequence ransforms ino he ensor produc { H 2 (S1 ) = H r (Z/2; H s (S 1 ; Z)) H r (Z/2; Z) H s (S 1 ; Z) r (Z/2; Z), for s = 0, 1 0, oherwise. Since, S 1 is a free Z/2 space we have ha EZ/2 Z/2 S 1 S 1 /(Z/2). The specral sequence (S 1 ) converges o H (EZ/2 Z/2 S 1 ; Z) H (S 1 /(Z/2); Z). Therefore, 3 (S1 ) (S 1 ) = 0 for all r + s 2 and so Index Z/2 (S 1 ; Z) = T. (2) 3.2. Index Z/2 (V 2 (R 3 ); Z) In order o describe he specral sequence (V 2 (R 3 )) recall ha he cohomology of he Siefel manifold is given by Z, s = 0, 3, H s (V 2 (R 3 ); Z) Z/2, s = 2, 0, oherwise. The E 2 -erm of he specral sequence (V 2 (R 3 )) can be now deermined in more deails H 2 (V r (Z/2; Z), s = 0, 3, 2(R 3 )) = H r (Z/2; H s (V 2 (R 3 ); Z)) = H r (Z/2; Z/2), s = 2, 0, oherwise.

5 P. Blagojević, A. Dimirijević Blagojević / Filoma 27:1 (2013), In paricular, he 1-row of he E 2 erm vanishes. Therefore, 0 T E 2,0 2 (V 2(R 3 )) E 2,0 (V 2 (R 3 )) and consequenly T Index Z/2 (V 2 (R 3 ); Z). (3) The relaion (3) will be enough o conclude he proof of Theorem 1.3. Neverheless, le us poin ou ha he specral sequence (V 2 (R 3 )) can be compued in all deails and i can be proved ha Index Z/2 (V 2 (R 3 ); Z) = T Proof of Theorem 1.3 According o Proposiion 2.1 in order o prove Theorem 1.3 we need o prove he non-exisence of a Z/2-equivarian map V 2 (R 3 ) S 1. The Z/2-acions on V 2 (R 3 ) and S 1 are assumed o be as inroduced in Secion 2. Assume he opposie, le f : V 2 (R 3 ) S 1 be a Z/2-equivarian map. The basic propery of he Fadell Husseini index heory (1) implies ha Index Z/2 (S 1 ; Z) Index Z/2 (V 2 (R 3 ); Z). This conradics he fac ha Index Z/2 (S 1 ; Z) = T, (2), and T Index Z/2 (V 2 (R 3 ); Z), (3). Thus, here can no be Z/2-equivarian map V 2 (R 3 ) S 1 and consequenly Theorem 1.3 holds. References [1] J. Arocha, J. Jerónimo-Casro, L. Monejano, E. Roldán-Pensado, On a conjecure of Grünbaum concerning pariions of convex ses, Periodica Mah. Hungar. 60 (2010) [2] P. Blagojević, G. Ziegler, The ideal-valued index for a dihedral group acion, and mass pariion by wo hyperplanes, Topology Appl. 158 (2011) [3] E. Fadell, S. Husseini, An ideal-valued cohomological index, heory wih applicaions o Borsuk-Ulam and Bourgin-Yang heorems, Ergod. Th. and Dynam. Sys. 8 (1988)